Linear Algebra

Explore systems of linear equations, identifying when solutions are unique, infinite, or nonexistent, and learn how plotting lines reveals consistency or inconsistency.

Learn to solve systems of equations by using augmented matrices, row operations, and transforming to row echelon form or reduced row echelon form with pivots and zeros.

Learn to transform matrices into reduced row echelon form using row operations, and understand pivots, zeros above and below, plus the distinction from row echelon form.

Apply Gauss-Jordan elimination to solve systems of equations using an augmented matrix and row operations, reaching the reduced row echelon form and identifying unique, infinite, or no solutions.

Learn to solve systems of equations by constructing the augmented matrix from coefficient matrix A and B, applying row operations, and solving word problems about stock shares.

this lecture shows how to fit a cubic f(x)=ax^3+bx^2+cx+d to three points, revealing an underdetermined system with a free parameter and solving it via augmented matrices and plotting in Python.

See how linear algebra underpins cryptography through Shamir's secret sharing, using polynomials and points to encode a secret so only a subset reconstructs it.

Analyze a network flow problem by modeling four junctions with inputs and outputs, forming equations and an augmented matrix to identify pivots and free variables.

Form a two-equation mixture model for five pounds at nine dollars per pound, yielding three and one third pounds of raisins and one and two thirds pounds of cashews.

This lecture analyzes the social security benefits crossover point between starting at 62 and waiting to 67, using algebra and linear algebra to model and compare totals.

Explore the matrix inverse for square matrices, understand existence, and learn inverse rules, transpose relationships, and identity matrices, and how to compute inverses via the 2x2 formula or augmented matrices.

Explore finding a matrix inverse using the inverse formula and row operations, and distinguish invertible from noninvertible matrices via determinant and pivot insights.

Apply matrix inverse rules to a two-by-two setup with B and C. Use (AB)^{-1} = B^{-1} A^{-1} and (A^{-1})^{-1} = A to determine A.

Explore the invertible matrix theorem for square matrices, proving equivalence of statements such as determinant nonzero, existence of an inverse, and pivot-based invertibility.

Explore the determinant of square matrices, including key properties, 2x2 and 3x3 computation via ad minus bc and the lattice method, and its relation to transpose, product, and inverse.

Explore how to compute determinants of larger matrices using permutation products and sign adjustments. Apply row operations to form upper triangular forms and relate the determinant to the original.

Use determinants to compute a polygon's area from vertex coordinates, noting orientation affects sign; take the absolute value to obtain the true area.

Use determinants to determine the orientation of triangle points, negative for clockwise, positive for counterclockwise, and zero for collinear, enabling real-time collision detection in computer graphics.

Explore vector spaces, the foundational linear spaces defined by closure under vector addition and scalar multiplication, with the axioms, zero vector, additive inverse, and distributive properties.

This lecture shows that many objects, like polynomials of degree two or less and two by three matrices, form vector spaces closed under addition and scalar multiplication, with zero vector.

Define subspaces as subsets of a vector space that contain the zero vector and are closed under addition and scalar multiplication, including lines through the origin as examples.

Examine subspaces of 2x2 matrices by testing zero inclusion, closure under addition and scalar multiplication for matrices that commute with A; show invertible 2x2 matrices are not a subspace.

View linear combinations as sums of scalar multiples of vectors and solve systems of equations by considering column vectors of A with coefficients x1, x2, …, equaling b.

Explore the span of a spanning set of vectors, via linear combinations, forming a subspace of the vector space, and test membership in that span using augmented matrices.

Explore the null space (kernel) of a matrix A as solutions to Ax = 0, using row reduction to row echelon form and pivot and free variables to describe span.

Examine the column space of a matrix, the set of all linear combinations of its columns, and determine if a vector B lies in this subspace by solving AX=B.

Linear independence means the only solution to a linear combination of vectors equaling zero uses zero coefficients; form a matrix from the vectors and examine the pivots.

Identify and remove redundant edges in a graph using linear independence and linear combinations, then apply reduced row echelon form to keep pivot edges and obtain linearly independent edge set.

Learn how a basis combines a spanning set with linear independence, and how pivot columns form a basis for subspaces, illustrated by planes in R2.

Examine the standard basis as the building blocks for vector spaces, showing how linearly independent vectors form bases in R2, P2, and 2 by 3 matrices.

Compute a basis for the span of four 2x2 matrices by stacking columns and applying row reduction, identifying pivot columns, and expressing dependent matrices as linear combinations.

The lecture shows how to find all 2x2 matrices that commute with a given 2x2 matrix, solving AB=BA to obtain a two-dimensional basis.

Explore linear transformations as vector space mappings represented by matrices, with T(x) = A x, and apply linearity conditions to distinguish linear from nonlinear mappings, including X = B.

Explore linear transformations in two dimensions by applying additivity and homogeneity tests to vectors in R^2, distinguishing linear mappings from nonlinear ones with concrete examples.

Explore a map from R^2 to R that outputs the length sqrt(x1^2 + x2^2) using Pythagoras and test linearity with u+v and scaling, showing it is not linear.

See how differentiating polynomials acts as a linear transformation from polynomials to polynomials, preserving additivity and scalar multiplication through F and G.

Learn how to identify linear transformations by expressing them as a matrix times the input vector, determine the matrix from coefficients, and understand input and output dimensions in examples.

Explore how two-dimensional geometric transformations use matrices to map vectors to new vectors, including reflections about the axes, scaling, and rotations via cosine and sine based matrices.

Apply rotation about a point in computer graphics by translating to the origin, rotating, and translating back, using homogeneous coordinates and matrix multiplication to combine transforms for many points.

Explore how dimension, rank, and nullity describe subspaces of matrices, reveal the rank-nullity relationship, and show how pivot columns and free variables determine column space and null space.

Explore isomorphisms between vector spaces via linear transformations, focusing on injective and surjective mappings that establish a one-to-one, onto correspondence.

Explore image classification in artificial intelligence by representing images as high-dimensional vectors, using training samples to form class centroids and classify new objects with Euclidean distance to the nearest centroid.

Explore coordinate systems by viewing linear systems and vectors as linear combinations of basis vectors, and learn how change of coordinates connects bases and coordinate vectors.

Discover how standard basis and other bases represent points, express coordinates as linear combinations, and perform change of basis using basis vectors.

Find the change of basis matrix from B to C and its inverse. Use augmented matrices and row operations to relate coordinates in B and C.

Explore how error-correcting codes use parity bits, generator and parity-check matrices, and modulo-two arithmetic to detect and correct single-bit errors in data transmission.

Learn how to rectify perspective in images by using a change of basis and homogeneous coordinates to map skewed points to a corrected trapezoid, demonstrating perspective correction.

Review dot products, norms, and unit vectors, then relate the angle between vectors to orthogonality. Extend the idea to inner products and orthogonal polynomials in higher dimensions.

learn how to identify orthogonal and orthonormal sets using dot products. verify pairwise orthogonality and convert to unit-length vectors by scaling, illustrating with example vectors.

Explore how to compute a plane's normal vector for ray tracing by using two independent vectors, the cross product, and determinants to obtain a vector orthogonal to the plane.

Learn how a basis for a subspace must span and be independent, and how orthogonal and orthonormal bases simplify math and solving linear systems.

Explore the orthogonal basis to decompose a vector into coefficients using dot products, enabling projection onto basis vectors and verifying linear independence.

Explore orthogonal matrices with orthonormal columns, where Q^T Q = I and Q^T = Q^{-1}. Preserve length under rotations and reflections.

Explore how orthogonal bases simplify expressing vectors and computing orthogonal projections, by decomposing a vector into projections onto V1 and V2 and recovering W from those components.

Explore orthogonal projections onto basis directions, decomposing vectors into components along v1 and v2. Visualize projections lying on the line spanned by the basis and using standard coordinates.

Project a vector onto a subspace spanned by an orthogonal basis by summing its projections onto each basis vector, as shown in the plane example.

Apply Gram-Schmidt to turn a linearly independent set into an orthogonal basis that spans the same subspace, by iteratively subtracting projections to make vectors orthogonal.

Explore the Gram-Schmidt process to turn independent vectors into an orthogonal basis, using projections and dot products, with examples in R2 and R3.

This lecture explains QR factorization of a matrix A into Q and R, with Q orthogonal and R upper triangular, including full and thin forms and connections to Gram-Schmidt.

Explore the least squares approach to approximate B with AX by projecting B onto A's column space, minimizing the residual and the sum of squared errors to obtain X*.

Derive the normal equations for least squares, showing that A^T A x = A^T b yields the best x, and note QR factorization as a more numerically stable alternative.

solve the least squares solution by forming normal equations A^T A x = A^T b and solving for x*. verify residual is orthogonal to column space using the two-norm.

Demonstrate least-squares solutions for consistent and inconsistent systems, and compare projecting onto a plane with orthogonal versus non-orthogonal bases, using the normal equations and Gram-Schmidt.

Explore linear regression with least squares, fitting a line to height and weight data by minimizing the sum of squared errors and solving via normal equations in matrix form.

Explore least-squares data modeling by fitting linear and quadratic models to data, estimating coefficients a0, a1, a2, and comparing fits via sum of squares and errors with Python plots.

Apply least squares to fit a circle to noisy x–y data by deriving the circle equation, forming a linear system, and solving for center and radius.

Fit periodic data with a trigonometric model, a linear combination of sine and cosine terms, using orthogonality and inner products for least-squares or interpolating fits on [0, 2π].

Define eigenvectors as nonzero vectors satisfying Av = lambda v, where lambda is an eigenvalue; use det(A - lambda I) = 0 to find lambdas and then corresponding eigenvectors.

Compute eigenvalues and eigenvectors for a 2x2 matrix by forming A minus lambda I, solving the characteristic polynomial, and deriving eigenvectors from the null space.

Explores special cases of eigenvalues and eigenvectors, including inverses, singular matrices with zero eigenvalues, and triangular matrices where eigenvalues are the diagonal entries, with examples.

Explore the trace of a square matrix as the sum of diagonal terms and its connection to eigenvalues, and use determinant as their product to derive the characteristic polynomial.

Apply principal component analysis to identify directions of maximum variance using eigenvectors and the covariance matrix, yielding orthogonal principal components and enabling dimensionality reduction for clustering and face classification.

Explore Markov chains through a two-state population model, using a migration matrix and stochastic matrices to derive steady-state distributions and eigenvector insights, with PageRank as a notable application.

Explore the singular value decomposition (SVD) of a matrix A, detailing U, Sigma, and V^T, the diagonal Sigma of singular values, and the full versus economy versions.

Explore singular value decomposition by constructing sigma from eigenvalues, forming orthogonal u and v, and handling nonzero and zero singular values for rectangular matrices.

Explore singular value decomposition and its link to the inverted matrix theorem, showing how rank equals non-zero singular values and illustrating full versus reduced SVD.

Explore general applications of the singular value decomposition, including solving overdetermined systems, computing pseudoinverses, and constructing low-rank approximations using leading singular values.

Apply singular value decomposition to classify images by projecting test samples onto class-specific subspaces built from training images and selecting the class with the smallest reconstruction error.

Explore image compression with singular value decomposition and low-rank approximations, reconstructing color images from a few dominant singular values in each color layer.